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Phenomenological evidence for the gluon content of η and η′
4 January 1996 PHYSICS
LETTERS
B
Physics Letters B 365 (1996) 367-376
Phenomenological
evidence for the gluon content of 33 and 7’
Patricia Ball a, J.-M. F&e b,‘, M. Tytgat b,2 a CERN, b Service de Physique
Theory Division,
Thiorique,
CH-1211
Gen&e
Universitc! Libre de Bruxelles,
Received 4 September Editor: R. Gatto
23, Switzerland CP 225, B-1050
Bruxelles,
Belgium
1995
Abstract We discuss the effect of v-7’ mixing and of axial current anomalies on various decay processes. We describe the 2y decays of 7 and v’, the radiative decays of low-lying vector mesons as well as the decays 71 --+ 37r, J/I) -+ w and J/JI + q’y in one unified framework, in which they are related to the electromagnetic and strong axial current anomalies, respectively. We also discuss briefly the enhancementof T,J’in D, decays and suggest a possible relation to gluon-mediated processes. The importance of the decays of the glueball candidates f~( 1500) and f~( 1590) into 77’ channels is also stressed.
In this paper we study the effect of q-q’ mixing and of the axial current anomaly on various processes. We show that a consistent picture arises for the radiative decays r] --f 2y and 7’ + 2y, for radiative decays of type V --) 777, B’ ---tVy of the lowestlying vector mesons V, as well as for v + 37~ and J/$ + v (7’) y. Whereas for all the other decays we follow closely the approach of Ref. [ 13, our description of the vector meson decays is based on their relation to the QED triangle anomaly. In Section 2 we thus first test our approach for the simpler cases of the decays V -+ T( K)y. In Section 3 we then establish notations for the q-r]’ system and determine the decay constants fo and fs as functions of the mixing angle from the 2y decays of 71and 7’. Section 4 is devoted to the comparison of our expectations for the widths of V -+ w, r]’ --t Vy with experiment, whereas Section 5 deals with the decay v -P 37r, which probes
the light quark content of r]. In Section 6 we then investigate the gluon content of 11and $, which determines the decays J/(cl 4 vy and J/$ --) v’y. It is in this channel that our approach yields a dependence of the involved matrix elements on the mixing angle that is significantly different from other models. We briefly comment on this in relation to the approach of Veneziano et al., Ref. [ 21, and close in Section 7 with some remarks on the important r6le of anomalies in other gluon-enriched channels like D, + q (q’) X and in glueball searches.
1.
’ Directeur de recherches * Aspirant du FNRS.
du FNRS.
2. The radiative decays of the lowest-lying vectormeson nonet are traditionally described in terms of magnetic moments of quarks, see e.g. [ 31, or of unknown couplings related by SUF( 3) symmetry [ 41. In Refs. [ 561 the width r(p --t ~7) was related via SUF( 3) arguments and vector-meson dominance to the radiative width of T. In this paper we relate the radiative decays of light vector mesons to light pseudoscalars, V -+ Py and P + Vy, directly to the
0370-2693/96/$12.00 Q 1996 Elsevier Science B.V. All rights reserved SSDI 0370-2693(95)01287-7
P. Ball et al./ Physics Letters B 365 (1996) 367-376
368
anomaly of the A W triangle diagram, where A stands for an axial-vector and V for a vector current. Our approach 3 both includes SIJr( 3) breaking effects and fixes the vertex couplings gvq as defined below 4 . Let us start by considering the correlation function i
d4xeiy2-r
(r(q
+@)I
TJ~“(X)J~=O(“(0)
IO)
s = E/.Wprrd94rF?r./=o(I ) c9:, 9;) 3
(1)
with the currents JEM = 5 2 uy,u -
H
$ dy,d
-
- f Sy,s,
The values of Fr., (0,O) are fixed by the QED triangle anomaly as F,,/=I (O,O> = -
1
Wf,'
F z-,/==o(O~O) = &?
7l
fT is defined
(0 IW,yd In-) = if,p,,
(4)
so that fT = 0.132 GeV from T --f ,w~. For the analogously defined kaon decay constant one finds fK = 0.160 GeV. Using their analytic properties, we can express these form factors by a dispersion relation in the momentum of the isospin current 5 : 03 0) = ;
.I 4(9)nI;
dsfImF,,,=~(~,(s,O).
k gp,ry + . . . ,
FT,ko(O,O) = $7,
+ ...
Here the dots stand for higher resonances and multiparticle contributions to the correlation function. In the following we assume vector meson dominance and thus neglect these contributions. Although this assumption may be criticized (and actually in Ref. [ 61 the continuum contributions were modelled by the perturbative spectral function above some threshold, but without taking into account the motion of the light quarks in the pion), it yields a description of the data that is good enough for our purposes6. The f v in (6) are the vector mesons’ leptonic decay constants defined by (7)
A denotes the helicity state of the meson. For ideal mixing the relevant currents are J$ = JA=‘, J;Lw= JL” and J$ = -Sy,s. In the following we include the deviation from ideal mixing by taking into account a mixing-angle 0~ = 40.3” for 4-w mixing7. The fv can be determined from the experimental decay rates [91 via r( V + e+e-)
2 f; = cv7jw -
w
(8)
with cv = {2/3,2/9 sin2 Bv, 2/9 cos2 19,) for V = {p’, w, 4). The experimental results are fP~~= (216 & 5) MeV,
(5) 3A
%,/=Itoto> =
(0 I Ji I V(P, A) ) = wfw, '*'(P)(3)
where the pion leptonic decay constant as
F,r,/-I (o)(O,
Saturating by the lowest-lying resonances we obtain
fw = (174f3)MeV, fb = (254% 3) MeV.
(9)
similar approach was proposed in Ref. [7].
4 The behaviour of the anomalous contribution has been studied quite extensively, in particular for large space-like momenta, and found to fall off asymptotically
like
1/
- q2
[ 8 1.
The results of
The charged mesons decay constants can be obtained from 7 + Vv, via
this section show, however, that in the physical sector, and for momenta up to the square of the $J mass, the full value of the
6 We are interested here in understanding how the gluon anomaly
anomaly gives sensible results. It was important for us to check this
affects decays involving n and 7’ and not so much in a detailed
fact in the p. o, rr, K sector, independently of further difficulties
fit of radiative vector meson decays.
associated with the strong anomaly. 5 Note that the dispersion relation needs no subtraction, and that the value Fn,t(O.O) is unambiguously fixed by the anomaly.
’ This value follows from the standard SUr( 3) breaking analysis; the ideal mixing angle is given by tanBv = I/& to Hv = 35.3O.
corresponding
369
P. Ball et al. / Physics Letters B 365 (1996) 367-376
Table 1 Theoretical and experimental values of the on-shell V-P electromagnetic vertex couplings defined in EQ. ( 12). For espy (th.) we give only the experimental errors coming from the decay constants fev P
V
RW (w.)
KW Cth.)
mp
= (0.73 f 0.07) GeV-’
(0.74 * 0.04) GeV-’
mp
= (0.73 i 0.07) GeV-’
(0.98fO.l3)GeV-’
- (2.58 f 0.04) GeV-’
(2.33 f 0.07) GeV-’
= (0.20*O.OI)GeV-’
(0.14+O.OI)GeV-
4f7rfP2 JfTrf@ 3m,
sin Bv + fi
4flrflo~2 -.?!%-
4f7rfcpp2
(&sin&
cos 0~
43 -cos&)
mK* 2fK’
mKL 4fK*
where vj is the appropriate CKM matrix element. We find f p* = ( 195 f 7) MeV, =
(226 f 28) MeV.
(11)
In view of the discrepancies between the measurements of the p decay constant from the charged and neutral sectors, we will use the average value fp = (205 !E 17) MeV in the following. Finally, we introduce vertex couplings gve, which are just the on-shell V-P electromagnetic form factors:
= -gvpy
(1.25fO.IS)GeV-’
( I .25 f 0.05) GeV-’
=
(0.62 k 0.08) GeV-’
(0.84 k 0.04) GeV-’
fKP2
(10)
fK .*
=
fKn2
epvpc,p;p;Ey)~‘.
(12)
The amplitude of the decay P -+ Vy or V + Py, depending on kinematics, is then obtained by contracting with the polarization vector of the photon and multiplying by &&. The decay rates read
(13)
In Table 1 we give both the theoretical and experimental values of the couplings gvb for decays involving pions or kaons. The agreement between theoretical and experimental values shows that the ground states indeed dominate the spectral functions in (5)) with corrections of order 10%. There are, however, three channels where the deviation from the experimental value is larger. The couplings in the p-r channels should be equal; those in the K*-K channel differ by a factor of two, which is not quite supported by the data. Since the couplings are related by Clebsch-Gordan coefficients, we see no possibility to reconcile the data with our predictions and leave this “anomaly” as an open question to the experimentalists. For 4 -+ ry the deviation is more than 10%. This decay, however, is in our approach completely due to 4-w non-ideal mixing and thus strongly suppressed compared to the other channels. It may also be influenced by other, usually negligible mechanisms, in particular p-w mixing, which we have neglected here, so that we consider the agreement with the experimental coupling as still satisfactory.
3. Having gained some control over the radiative decays of vectors into 7~and K, we now turn to the vv’ system, which is our central point of interest in this study. Numerous possibilities, of which we can only quote a few [ 101, have been suggested to describe this system, with or without explicit mixing with extra “glueball” states. The discussion of all these ap-
P. Ball er al. /Physics Letters B 365 (19961367-376
370
proaches goes far beyond the scope of this letter, and we will content ourselves with shortly commenting in Section 6 on the elegant approach by Veneziano et al., Ref. [2]. For the time being, however, we focus on the more simple and pragmatic approach proposed in Ref. [ 1I. Following the Particle Data Group conventions, which differ from the ones originally used in Ref. [ 11, we reproduce here the key formulas in the current notation. In terms of the mixing angle 8 and their singlet and octet components 70 and 78, respectively, the physical states are decomposed as
As in Ref. [ 11, our essential assumption is that the 11and d quark masses can be neglected in ( 15). This yields the following simple expressions for the matrix element of the strong anomaly over the vacuum and rl(rl’):
17) = (ns) cos 13- ITO) sine,
We thus have three parameters to fix: fo, j”s and 0. Two of them can be determined from the radiative widths of v and v’, which are given by
Jr]‘) = 1~~) sine + 1~~)cos e.
(14)
Traditional SUF (3) -based analyses suggest either 0 N -10’ or B N -23” from the quadratic and linear versions, respectively, of the Gell-Mann-Okubo mass formula, whereas we leave 8 as an open parameter, at least for the time being. Analogously to the pion decay constant f,,., Eq. (4), we define the decay constants * fs and fe as couplings of 7s and 70, respectively, to the divergence of the relevant axial-vector currents (where fs = fr # fo in the SUF( 3) limit):
(17)
m;
I(77--r2y)=----(y 961r”
r(7f --f2y)
rn$ 2 = cy 96~’
sine -+ f8
(19) As in Ref. [ I ] we solve these relations for the decay constants and thus get fo and fg as functions of the mixing angle. As experimental input we use [9] I(q + 2~) = (0.51 f 0.026) keV, I( 7’ --f 2~) = (4.53 & 0.59) keV,
where G$, is the gluonic field-strength tensor and z;* =&2 fi,,pr,GApaits dual. Defining as in Ref. [ 1 ] thzYinterpolating fields of r] and 77’as linear combinations of the axial-vector divergences, we thus have (OlJ,A~j~)
-m~.fscos&
(OlJ,A,“Iq)
=-mtfcsin8,
(0 I c?,A~ ( 7’) = rn$ fg sine, (Ola,ArI+)
=mi,focosO.
(16)
where according to the suggestion in the full listings part of the Review of Particle Properties (Ref. [ 91, p. 145 1) we have only retained the more recent data. In Fig. 1, fo/fTand fs/ ffl are plotted as functions of the mixing angle (solid lines) together with their experimental error (dashed lines). We find thus fg/fT2 1 for 0 less than N”- 15’ as expected from chiral perturbation theory. The precise value [ 111 fs/fr= 1.25, however, corresponds to a rather large’ mixing 0 = -21.3”. 9 In fact, Ref.
fo
[ 121
supplements the more traditional approaches
to the 2y decays of 7 and 7 contributions,
x Note that by virtue of the “hard” non-conservation of A:,
(20)
by taking into account continuum
which just cancel the effect of the large S&(3)
I .25.
breaking ratio fg/fT
=
is scale-dependent as described in detail in the first reference of
with the experimental
input data of Eq. (20).
The result is then 19= -(
17 f
2)’
[2]; we will assume here that the momentum dependence can be
the following, this value is in excellent agreement with the results
neglected.
we find from the investigation of other decay channels.
As we shall see in
P. Ball et al./ Physics Letters B 365 (1996) 367-376
371
Table 2 Theoretical formulas and experimental values of the on-shell V-q, 7’ electromagnetic vertex couplings defined in Eq. ( 12). In Fig. 2 the theoretical values are plotted as functions of 0. We use 8v - 40.3’ for the 4-w mixing angle. Experimental values are taken from [9] P
V
77
P
gvpy in units -
W
gwy (exp.)
fvp2
-A
4fx
cosH -
J
J
31,
- -
8 fo
3 I -costi + J5 sin0 4fu 8 fo
P
(I.31 &O.l2)GeV-’
rl’
w
cos 0 sin 0 sin Hv (v5cos.9~ - sinfIv) - ___ 4fx 2&I cos 8 sin#v + @ (JzcosBv - sin&)
‘1
4J
--
‘I’
4
-
w
4fu
Mfo
cos 0 4fx
cos 0
2dfo
(cos&~ + &sin&)
COSSV- !!Z
4fu
2.5 2 1.5 1 0.5 -25
-20
-15
-10
-5
sin 0 - 2~zfo
cos&
(cost+ + &sin&)
3
0
( 1.85 i 0.34) GeV-’
sin0
0
I9 Fig. I. fo/fr and fg/fv,determined from q -+ 2y and 8’ --t 2y, as functions of the mixing angle. Having thus fixed fc and fs, we continue in the next section with the electromagnetic properties of 7 and v’, i.e. the vector meson decays, consider then in Section 5 their light quark matrix elements, to finish in Section 6 with their glue content.
4. We are now in a position to deal with the radiative vector meson decays involving 7 or 7’. In Table 2 we list the decay channels together with the theoretical formulas for the couplings go+ and their experimental values. In Fig. 2 we then plot the theoretical and experimental values as functions of the pseudoscalar mixing angle. Since in our approach continuum contri-
(0.60fO.l5)GeV(0.45 L!Z 0.06) GeV-’ (0.70 f 0.03) GeV-’
< 1.85 GeV-’
butions are neglected, the couplings should be slightly overestimated by about 10 to 15%, as one may also infer from the cleanest channel w --t 7ry investigated in Section 2 (cf. Table 1). A look at Fig. 2(b) shows that this is not the case for 7’ --f py. In that channel, however, we expect a contamination of the experimental value from non-resonant VJTfinal states, which are produced by a completely different mechanism, which is related to the box-anomaly diagram (cf. e.g. [ 131). For p + vy and o --) r]y, Figs. 2(a) and (c), the experimental errors are too large to allow any serious restriction on 8. Within the error bars, however, our predictions agree with the experimental values. gwtljy, on the other hand, is very sensitive to the 4-w mixing angle; for instance, reducing 8~ by 1.5” brings the coupling within 20% agreement with the experimental data, i.e. just what we would expect. As seen from the figures, the most stringent constraint on 0 currently stems from the decay I$ -+ qy. It is largely independent of the 4-w mixing angle and, as it will turn out below, only marginally consistent with the range of mixing angles found from other processes (remember, however, that we use 1 (+ experimental errors, and do not take into account theoretical uncertainties). The issue should be clarified when experimental errors eventually shrink, and in particular we look forward to a measurement of the 4 ---f $y mode, for which our prediction is only a factor 2 below the current experimental upper bound.
P. Ball et al./Physics
312
Letters B 365 (1996) 367-376
(4
(b) 1.6
-i
2
El.
F
cn
2.2 2 1.8 1.6 1.4 1.2 1
I I I I(/
I,,,,,,,,,,,,
I,,,
/,,,I
_ d $
1.5
5
1.3
1.4
=; 1.2 cn 1.1 -25 -20 -15 -10 -5 8
0
.-‘_ .’ .-
_._._._._.-._._._._.-.r’-._._., ----_____.---
.-.-
._._._.-.
- .-.-._._._.
1 kl I
_.__,“._ .-
F _‘-__
_.---__ )
I/
.--
--____.----
I
I
I
I 1
I
I
I I I 1
I,
1 , 1
I
(C) 0.80 0.75 0.70 0.65 0.60 0.55 0.50 0.45 0.40
I I
, I I I I,,
I I I,
I,
I I{
I
//
Id
-25 -20 -15 -10 -5 0
0 (d)
I / , , I
.-.-.-.-._._._._._.-.-.~.~.~
.-.-.-._._._._._._.-.-.-._.-
I1
._._._._._.-._._____.-.-._._ /II I I I I,, II1 I I I I I I
/ /
I
I I I
-25 -20 -15 -10 -5
I
0
“~‘1”~“1’~‘~“~‘~~11”“’
-25 -20 -15 -10 -5 0
0 (f)
1 0.9
0.8 0.7 0.6 0.5
-25 -20 -15 -10 -5
0
-25 -20 -15 -10 -5
0
0 Fig. 2. Theoretical and experimental values of the V-q, 7’ electromagnetic couplings as functions of the pseudoscalar mixing angle. Solid lines: theoretical predictions according to Table 2. Short-dashed lines: errors of the theoretical couplings coming from the experimental errors of fo and fs. Dashed-dotted lines: experimental couplings including errors (for g4,r,r, p lotted in (f), there exists only an experimental upper bound).
313
P. Ball et al. /Physics L.&fen B 365 (1996) 367-376
5. Let us now turn to the investigation of other matrix elements of 7 and q’ that can be probed in our approach. We start with the light quark matrix elements, which determine the decay 77 --t 37~.As in Ref. [ 1] we find lo : 1 focos0 - JZfssin8 (“‘W5u”)=
x
- Jz
fifocos8+fssin8
5
0.6
: 3-
0.5
7
0.4
g
0.3
k
0.2
(21)
(Ol~Y54V)~
-_-_-_-_-_~_-.~_ _._._._._.~._ b .-.-___-._(_-
x f~cos*t? -f:
4-I
\
11”“““““’ -25 -20
1,
-15
.
‘b”l”““‘lq -10 -5
_$ 0
Fig. 3. Theoretical (solid line) and experimental values (dashed-dotted lines, including experimental error) of the decay rate T(r) + 3?r”) as function of the pseudoscalar mixing angle. The short dashes give the theoretical error of the rate, which is dominated by the error of the quark mass ratio.
mi(m, - 3m,)*&(E!$)* focos8-JZfssint9 ( JzfocosB
* *
0
I-(?? + 37rO)
45
_L-.-.-.-.-.-.-
1
which yields the decay rate
=s
_C-.a.-.-_-.-.-.
, \ \
+ fg sin0 >
* ’
(22)
with a kinematical factor S,, = 0.86. The most crucial ingredient in that formula is the ratio of quark masses r = (md - m,)/m,. There exist several possibilities to fix r from next-to-leading order calculations in chiral perturbation theory, as discussed in [ 141, for example. One of them is based essentially on the analysis of r] + 37r in Ref. [ 151. Since in that study 71is necessarily treated as a Goldstone boson, the value of r obtained that way is inconsistent with our analysis. For similar reasons, we also refrain from taking into account r determined from r( (//’ + J/h) /r( I,Q --+ J/@r) . We thus rely on the determination of r from pseudoscalar mass relations, which according to Ref. [ 141 yield r = 0.030 f 0.005. In Fig. 3 we plot the decay rate as a function of 8. Although the agreement with the experimental rate is very good for 0 x -17“, the theoretical errors are large and dominated by the uncertainty in the quark mass ratio. Actually the true theoretical error may be even larger and the rate enhanced by final state interactions [ 14,151. In Ref. [ I] we had also considered the decay 7’ -+ 31r. We have, however, since argued [ 161 that this channel is considerably more complex and also receives contributions from the leading decay mode 7’ ----f~TTT through 9-r mixing. While this study is of great interest in itself and while a measurement of loEq. (21 ) differs slightly from the corresponding Eq. ( 19) in Ref. [ I], where we had made the simplification fo/fs N i
the charged mode v’ 4 ~T+T-T’ could shed considerable light on the mechanism underlying 7’ --t ~ITV, it is of little help in determining the value of 8. 6. We finally turn to the most important issue concerning the 7-77’ system, namely the glue connection. Following the suggestion of Novikov et al., Ref. [ 171, the glue matrix elements are tested in the ratio of the decay rates r(J/$ -+ v(+)y):
=
(O/G&') (OIG~IT)
*
(1-+/m:/,)” (1 -mG/m:/,)”
(23)
Before turning to numerics, let us first comment shortly on the theoretical accuracy of the above equation. Actually the decay rates were calculated in exactly the same approximation we used in deriving the gvh given in the tables, namely dominance of the ground state and neglection of continuum contributions to the dispersion relations. It was also pointed out in Ref. [ 181 that the decay mechanism via the strong anomaly dominates only due to the smallness of the c quark mass and may be non-effective already in Y decays, for which unfortunately no experimental data exist to date. It is for these reasons that we prefer to consider the ratio RJI, instead of the two decay rates separately: both radiative and continuum corrections are expected to cancel to some extent in RJ/$.
374
P. Ball et al. / Physics Letters B 365 (I 996) 367-376
these initial fields. While various procedures are possible (like introducing some “glue” Q in the definition of v’, as we did above when using the total divergences as interpolating fields), the choice of basis is essentially free, and the authors of Ref. [ 21 preferred to redefine the Q field, introducing G as below ” . This leads to (we follow closely the notation of [ 2 ] ):
-25
-20
-15
-10
-5
0
0
G=Q-
Fig. 4. RJI$ = F( J/q5 -+ $y)/T( J/I+ ---) 7~) as function of the mixing angle 8. Solid line: RJ/, according to Eqs. ( 17) and (23) including experimental errors of fu and fs (short dashes). Long-dashed line: the ndive prediction (24). Dashed-dotted lines: experimental value with errors.
In Fig. 4 we plot R,/+ according to Eqs. (17) and (23) and also the naive prediction
0-
m~+&d3
RJl* =Cot2* (1 _ m;/m;,,)”
(24)
as functions of the mixing angle. As experimental input we use RT$ = 5.0 f 0.6 [9]. Our prediction (23) is rather sensitive to the mixing angle. If we estimate the remaining theoretical uncertainty in (23) as N lo%, we obtain 0 = -( 17.3 f 1.3)‘. It is clearly seen, on the other hand, that the naive curve is less consistent with the data and requires a very large mixing angle. To understand the departure of our formula from the ndive prediction, we briefly evoke the elegant, but heavier formulation by Veneziano and collaborators [2]. These authors deal with a simplified model, where r]s-ve mixing is neglected. To apply their approach to the physical situation requires the introduction of three initial fields (instead of two in our approach) :
(25) The definition of interpolating fields corresponding to physical particles requires the orthonormalisation of
(26)
where the coefficients bi and Wij can be determined in terms of two-point functions. Expressing the relevant Ward-Takahashi identities in terms of the effective action and taking two derivatives with respect to the gluon fields (g, with indices omitted below), we obtain for the (non-anomalous) octet axial current
=0
(27)
with X a computable coefficient. Clearly, if the proper vertex between G and two gluons could be neglected (in the way the similar coupling between G and two photons is legitimately neglected in Ref. [ 2]), we would recover the ndive result (24). This step is, however, not permitted since the proper vertex between C (whatever its physical realization) and two gluons is precisely expected to be large. It should thus not be surprising that we have to depart from Eq. (24), which is difficult to reconcile with the data, as already noted by Gilman and Kauffman in Ref. [ IO]. We will not delve deeper into this approach, except to mention that it is a clear generalization of ours (in our case the weight of glue in 7’ is directly fixed by our definition of its interpolating field, instead of resulting from a further diagonalization between the three states). 7. Having a rather comprehensive view of the 17 and 77’system and its connection with glue, we now address other promising channels for future investigations, to wit D,y decays. Here two processes compete ‘I Note that the question if G is to be identified with any physical state, a glueball for instance, is left open in Ref. [21.
P. Bail et al./ Physics Letters B 365 (19%) 367-376
to produce 7;1 (v’) X final states: the first one is the decay c + s, leading to an ss pair, which hadronizes to r] or v’, while the second one can be described as CS annihilation into W and two gluons. It is suppressed by the Zweig rule, but may in the present case gain importance through the large (0 )GG (q(7'))matrix elements. We should thus expect both an enhancement of the (~7+ 7$)X modes (unless the interference is destructive) and a large q//v ratio, once the rates are corrected for phase-space and final state interactions. If we ignore for the moment final state interactions, the simplest way to evaluate a possible enhancement is by direct comparison to similar DO decays. For instance the partial width for DO -+ K-p+ (to which only the first diagram contributes) would, in the absence of the glue contribution, be expected to be of the same order (after phase-space corrections) as the --) q’p. Instead we find sumD,-+r]p+D,
375
to be enhanced by an extra factor of up to three when compared with the corresponding DO decay amplitudes, even after including final state interactions; and the global fit done in Ref. 1211 for all non-leptonic D and D, decays including final state interactions, W exchange and annihilation contributions fails to explain the large branching ratios of D, into ~7’71and q’p. The enhancement needed is, however, well in line with our expectations from the extra Zweig-suppressed (gluonmediated) amplitudes. Very recently, the CLEO collaboration has measured the theoretically cleanest semileptonic channel and finds [22] r( DS + T’ev) = 0.35 f 0.09 f 0.07. I(D, ---fTV)
(32)
Defining the relevant form factors as (rlI~~,cID.~)=f:(q~)(~o,+~~),
I(DS -+ %+)(Pq)-3 -tI(D, + 71P)(P71)-3 I(Do --f K-p) (PK)-3
+ fL(#)
2
x!z?i
( mDC,)
=
4.2 f 1.O,
(28)
while for the branching ratios we find =4.5& 1.5,
(29)
which indicates both an enhancement of the decay rate and a large ratio for these important decay modes (of the order of 10% each). The situation is unfortunately less clear in the T associated modes, where the comparison with DO now gives + T/T) (P,~) -1 + ua
w. = 1.6 f 0.4,
---) K-T)
--) VT) (P,)-’
m-l
(30)
while (31) More sophisticated analyses of non-leptonic D, decays were the subject of several studies, e.g. [ 19-211, which, however, mostly relied on the ndive picture of v-$ mixing without taking into account the anomaly. In Ref. [ 201 it was pointed out that in order to account for the data the non-leptonic rates involving 7’ need
(33)
with q = pD, - p,, and analogously J$, CLEO’s measurement yields
f$(O) = 1.14*0.17f0.13, f:(o)
-
-3
r(D,
(PO, - P1I)p )
(34)
where we assumed monopole form factors I2 with a pole at q’ = mZ,.. The ndive mixing model, on the other hand, predikts cos8 - &sin0 fh2) = fP(q2> sin0 + Jicosd’
(35)
which translates into 8 = -( 13.5 f 4.7)’ and is only marginally consistent with the determination from 77-+ 2Y [121. As a temporary conclusion to this section, we would like to add that, while this sector is clearly difficult to investigate both theoretically and experimentally, it could prove very fruitful for our understanding of the role of anomalies and the importance of gluons. There is also the strong suggestion that D, decays, not unlike the J/$ radiative decays, constitute a “gluerich” channel, where search for possible glueball states ‘*We
do not expect the anomalous contribution to change drasti-
cally the y* dependence of the form factors, which in D -+ Kev is a monopole experimentally.
376
P. Ball et al./Physics Letters B 365 (1996) 367-376
should be considered. We hope to come back to that issue in future work. Let us finally remark that apart from understanding the structure of the 77 and 7’ particles, the size of their connection to glue is of importance to investigate possible glueball candidates [23]. In particular, the candidates f0 ( 1500) from Crystal Barrel [ 241 and fc ( 1590) [ 251 are seen to decay more frequently into vr]’ than into 77~ (after phase-space correction), a ratio which we expect to be given by
where pt$,, are the respective centre-of-mass momenta. This relation is, however, highly momentum dependent, since we are close to the threshold, and obviously needs to be integrated over the particle width. In conclusion, the above study gives a reasonably consistent description of the 11-v system for values of the mixing angle /3between -20 and - 17 degrees, taking into account anomalies, as tested from various channels, respectively depicting eletromagnetic properties, light quark and gluon content. Our approach will be tested by a general improvement of the experimental measurements, including the V -+ Py and P -+ Vy rates, and in particular by the observation of 4 --+ r]‘y. The investigation of “glue-rich” channels, namely the traditional radiative decays of J/J/, but also, as advocated here, the D,$ decays, will also yield a better understanding of this system and at the same time help in the search for glueballs, which might solve the puzzle. We thank C. Amsler, M. Benayoun, X.Y. Pham, M. Shifman and G. Veneziano for interesting discussions. P.B. gratefully acknowledges the hospitality of the LPTHE of Universite Paris VII, where part of this work was done. J.-M.F. also acknowledges discussions with 1.1.Bigi. This work was supported in part by the European Network Flavourdynamics (ref. chrx-ct930132) and NATO grant (ref. CRG920611).
References [ I ] R. Akhoury and J.-M. F&e,
[2] GM. Shore and G. Veneziano, 3, 23; S. Narison, GM. 433 ( 1995) 209. [3] [4] [53 [6]
( 1989) 258.
Shore and G. Veneziano,
Nucl. Phys. B
J. Bartelski and S. Tatur, Phys. Len. B 289 ( 1992) 429. M. Benayoun et al., 2. Phys. C 65 ( 1995) 399. M.S. Chanowitz, Phys. Rev. Lett. 44 ( 1980) 59. M.A. Shifman and M.I. Vysotsky, Z. Phys. C IO ( 1981) 131.
[7] A.N. lvanov (1980) 275.
and V.M. Shekhter,
Sov. J. Nucl.
Phys.
31
[S] G.P. Lepage and S.J. Brodsky, Phys. Lett. B 87 ( 1979) 359; Phys. Rev. D 22 ( 1980) 2157; S. Narison, G.M. Shore and G. Veneziano, Nucl. Phys. B 391 (1993) 69; G.M. Shore and G. Veneziano, Mod. Phys. Lett. A 8 ( 1993) 373; J.-M. Gerard and T. Lahna, Phys. Lett. B 356 ( 1995) 381. [9] L. Montanet et al., Review of Particle Properties, Phys. Rev. D 50 (1994) 1173. [IO] F.J. Gilman and R. Kauffman, Phys. Rev. D 36 ( 1987) 2761; P. Ko and T.N. Truong, Phys. Rev. D 42 ( 1990) 2419; A.V. Kisselev and V.A. Petrov, 2. Phys. C 58 (1993) 595. [I I ] J.F. Donoghue, B.R. Holstein and Y.C.R. Lin, Phys. Rev. Len. 55 ( 1985) 2766. [ 121 T.N. Pham, Phys. Lett. B 246 ( 1990) 175. [ 131 M. Benayoun et al., Z. Phys. C 58 (1993) 31. [ 141 J.F. Donoghue and D. Wyler, Phys. Rev. D 45 ( 1992) 892; J.F. Donoghue, B.R. Holstein and D. Wyler, Phys. Rev. Lett. 69 ( 1992) 3444. [ 151 J. Gasser and H. Leutwyler, Nucl. Phys. B 250 ( 1985) 539. [ 161 P. Castoldi and J.-M. F&e, Z. Phys. C 40 ( 1988) 283. [ 171 V.A. Novikov et al., Nucl. Phys. B 165 (1980) 55. [ 181 V.N. Baier and A.G. Grozin, Nucl. Phys. B 192 ( 1981) 476; V.L. Chemyak and A.R. Zhitnitsky, Phys. Rep. 1I2 ( 1984) 173. [ I9 J A.N. Kamal and R. Sinha, Phys. Rev. D 36 ( 1987) 35 IO; G.R. Dzhibuti, V.G. Kartvelishvili and Sh.M. Esakiya, Phys. At. Nucl. 56 ( 1993) 560; R.C. Verma, A.N. Kamal and M.P. Khanna, Z. Phys. C 65 ( 1995) 255; A.N. Kamal and A.B. Santra, preprint ALBERTA-THY- I-95 (hep-ph/9501246). [ 201 T.N. Pham, Phys. Rev. D 46 ( 1992) 2080. [21] E Buccella et al., Phys. Rev. D 51 (1995) 3478. [22] G. Brandenburg et al. (CLEO Co]].), preprint CLNS-95135I (hep-ex/9508009). [23] S.S. Gershtein, A.K. Likhoded and Yu.D. Prokoshkin, Z. Phys. C 24 ( 1984) 305; C. Amsler and F.E. Close, preprint RAL-TR-95-003 (hepphl9507326). [ 241 C. Amsler et al., Phys. Lett. B 340 (1994) 259. [ 251 D. Aide et al., Phys. Lett. B 201 f 1988) 160, and references therein.
Phys. Lett. B 220
Nucl. Phys. B 381 (1992)
LETTERS
B
Physics Letters B 365 (1996) 367-376
Phenomenological
evidence for the gluon content of 33 and 7’
Patricia Ball a, J.-M. F&e b,‘, M. Tytgat b,2 a CERN, b Service de Physique
Theory Division,
Thiorique,
CH-1211
Gen&e
Universitc! Libre de Bruxelles,
Received 4 September Editor: R. Gatto
23, Switzerland CP 225, B-1050
Bruxelles,
Belgium
1995
Abstract We discuss the effect of v-7’ mixing and of axial current anomalies on various decay processes. We describe the 2y decays of 7 and v’, the radiative decays of low-lying vector mesons as well as the decays 71 --+ 37r, J/I) -+ w and J/JI + q’y in one unified framework, in which they are related to the electromagnetic and strong axial current anomalies, respectively. We also discuss briefly the enhancementof T,J’in D, decays and suggest a possible relation to gluon-mediated processes. The importance of the decays of the glueball candidates f~( 1500) and f~( 1590) into 77’ channels is also stressed.
In this paper we study the effect of q-q’ mixing and of the axial current anomaly on various processes. We show that a consistent picture arises for the radiative decays r] --f 2y and 7’ + 2y, for radiative decays of type V --) 777, B’ ---tVy of the lowestlying vector mesons V, as well as for v + 37~ and J/$ + v (7’) y. Whereas for all the other decays we follow closely the approach of Ref. [ 13, our description of the vector meson decays is based on their relation to the QED triangle anomaly. In Section 2 we thus first test our approach for the simpler cases of the decays V -+ T( K)y. In Section 3 we then establish notations for the q-r]’ system and determine the decay constants fo and fs as functions of the mixing angle from the 2y decays of 71and 7’. Section 4 is devoted to the comparison of our expectations for the widths of V -+ w, r]’ --t Vy with experiment, whereas Section 5 deals with the decay v -P 37r, which probes
the light quark content of r]. In Section 6 we then investigate the gluon content of 11and $, which determines the decays J/(cl 4 vy and J/$ --) v’y. It is in this channel that our approach yields a dependence of the involved matrix elements on the mixing angle that is significantly different from other models. We briefly comment on this in relation to the approach of Veneziano et al., Ref. [ 21, and close in Section 7 with some remarks on the important r6le of anomalies in other gluon-enriched channels like D, + q (q’) X and in glueball searches.
1.
’ Directeur de recherches * Aspirant du FNRS.
du FNRS.
2. The radiative decays of the lowest-lying vectormeson nonet are traditionally described in terms of magnetic moments of quarks, see e.g. [ 31, or of unknown couplings related by SUF( 3) symmetry [ 41. In Refs. [ 561 the width r(p --t ~7) was related via SUF( 3) arguments and vector-meson dominance to the radiative width of T. In this paper we relate the radiative decays of light vector mesons to light pseudoscalars, V -+ Py and P + Vy, directly to the
0370-2693/96/$12.00 Q 1996 Elsevier Science B.V. All rights reserved SSDI 0370-2693(95)01287-7
P. Ball et al./ Physics Letters B 365 (1996) 367-376
368
anomaly of the A W triangle diagram, where A stands for an axial-vector and V for a vector current. Our approach 3 both includes SIJr( 3) breaking effects and fixes the vertex couplings gvq as defined below 4 . Let us start by considering the correlation function i
d4xeiy2-r
(r(q
+@)I
TJ~“(X)J~=O(“(0)
IO)
s = E/.Wprrd94rF?r./=o(I ) c9:, 9;) 3
(1)
with the currents JEM = 5 2 uy,u -
H
$ dy,d
-
- f Sy,s,
The values of Fr., (0,O) are fixed by the QED triangle anomaly as F,,/=I (O,O> = -
1
Wf,'
F z-,/==o(O~O) = &?
7l
fT is defined
(0 IW,yd In-) = if,p,,
(4)
so that fT = 0.132 GeV from T --f ,w~. For the analogously defined kaon decay constant one finds fK = 0.160 GeV. Using their analytic properties, we can express these form factors by a dispersion relation in the momentum of the isospin current 5 : 03 0) = ;
.I 4(9)nI;
dsfImF,,,=~(~,(s,O).
k gp,ry + . . . ,
FT,ko(O,O) = $7,
+ ...
Here the dots stand for higher resonances and multiparticle contributions to the correlation function. In the following we assume vector meson dominance and thus neglect these contributions. Although this assumption may be criticized (and actually in Ref. [ 61 the continuum contributions were modelled by the perturbative spectral function above some threshold, but without taking into account the motion of the light quarks in the pion), it yields a description of the data that is good enough for our purposes6. The f v in (6) are the vector mesons’ leptonic decay constants defined by (7)
A denotes the helicity state of the meson. For ideal mixing the relevant currents are J$ = JA=‘, J;Lw= JL” and J$ = -Sy,s. In the following we include the deviation from ideal mixing by taking into account a mixing-angle 0~ = 40.3” for 4-w mixing7. The fv can be determined from the experimental decay rates [91 via r( V + e+e-)
2 f; = cv7jw -
w
(8)
with cv = {2/3,2/9 sin2 Bv, 2/9 cos2 19,) for V = {p’, w, 4). The experimental results are fP~~= (216 & 5) MeV,
(5) 3A
%,/=Itoto> =
(0 I Ji I V(P, A) ) = wfw, '*'(P)(3)
where the pion leptonic decay constant as
F,r,/-I (o)(O,
Saturating by the lowest-lying resonances we obtain
fw = (174f3)MeV, fb = (254% 3) MeV.
(9)
similar approach was proposed in Ref. [7].
4 The behaviour of the anomalous contribution has been studied quite extensively, in particular for large space-like momenta, and found to fall off asymptotically
like
1/
- q2
[ 8 1.
The results of
The charged mesons decay constants can be obtained from 7 + Vv, via
this section show, however, that in the physical sector, and for momenta up to the square of the $J mass, the full value of the
6 We are interested here in understanding how the gluon anomaly
anomaly gives sensible results. It was important for us to check this
affects decays involving n and 7’ and not so much in a detailed
fact in the p. o, rr, K sector, independently of further difficulties
fit of radiative vector meson decays.
associated with the strong anomaly. 5 Note that the dispersion relation needs no subtraction, and that the value Fn,t(O.O) is unambiguously fixed by the anomaly.
’ This value follows from the standard SUr( 3) breaking analysis; the ideal mixing angle is given by tanBv = I/& to Hv = 35.3O.
corresponding
369
P. Ball et al. / Physics Letters B 365 (1996) 367-376
Table 1 Theoretical and experimental values of the on-shell V-P electromagnetic vertex couplings defined in EQ. ( 12). For espy (th.) we give only the experimental errors coming from the decay constants fev P
V
RW (w.)
KW Cth.)
mp
= (0.73 f 0.07) GeV-’
(0.74 * 0.04) GeV-’
mp
= (0.73 i 0.07) GeV-’
(0.98fO.l3)GeV-’
- (2.58 f 0.04) GeV-’
(2.33 f 0.07) GeV-’
= (0.20*O.OI)GeV-’
(0.14+O.OI)GeV-
4f7rfP2 JfTrf@ 3m,
sin Bv + fi
4flrflo~2 -.?!%-
4f7rfcpp2
(&sin&
cos 0~
43 -cos&)
mK* 2fK’
mKL 4fK*
where vj is the appropriate CKM matrix element. We find f p* = ( 195 f 7) MeV, =
(226 f 28) MeV.
(11)
In view of the discrepancies between the measurements of the p decay constant from the charged and neutral sectors, we will use the average value fp = (205 !E 17) MeV in the following. Finally, we introduce vertex couplings gve, which are just the on-shell V-P electromagnetic form factors:
= -gvpy
(1.25fO.IS)GeV-’
( I .25 f 0.05) GeV-’
=
(0.62 k 0.08) GeV-’
(0.84 k 0.04) GeV-’
fKP2
(10)
fK .*
=
fKn2
epvpc,p;p;Ey)~‘.
(12)
The amplitude of the decay P -+ Vy or V + Py, depending on kinematics, is then obtained by contracting with the polarization vector of the photon and multiplying by &&. The decay rates read
(13)
In Table 1 we give both the theoretical and experimental values of the couplings gvb for decays involving pions or kaons. The agreement between theoretical and experimental values shows that the ground states indeed dominate the spectral functions in (5)) with corrections of order 10%. There are, however, three channels where the deviation from the experimental value is larger. The couplings in the p-r channels should be equal; those in the K*-K channel differ by a factor of two, which is not quite supported by the data. Since the couplings are related by Clebsch-Gordan coefficients, we see no possibility to reconcile the data with our predictions and leave this “anomaly” as an open question to the experimentalists. For 4 -+ ry the deviation is more than 10%. This decay, however, is in our approach completely due to 4-w non-ideal mixing and thus strongly suppressed compared to the other channels. It may also be influenced by other, usually negligible mechanisms, in particular p-w mixing, which we have neglected here, so that we consider the agreement with the experimental coupling as still satisfactory.
3. Having gained some control over the radiative decays of vectors into 7~and K, we now turn to the vv’ system, which is our central point of interest in this study. Numerous possibilities, of which we can only quote a few [ 101, have been suggested to describe this system, with or without explicit mixing with extra “glueball” states. The discussion of all these ap-
P. Ball er al. /Physics Letters B 365 (19961367-376
370
proaches goes far beyond the scope of this letter, and we will content ourselves with shortly commenting in Section 6 on the elegant approach by Veneziano et al., Ref. [2]. For the time being, however, we focus on the more simple and pragmatic approach proposed in Ref. [ 1I. Following the Particle Data Group conventions, which differ from the ones originally used in Ref. [ 11, we reproduce here the key formulas in the current notation. In terms of the mixing angle 8 and their singlet and octet components 70 and 78, respectively, the physical states are decomposed as
As in Ref. [ 11, our essential assumption is that the 11and d quark masses can be neglected in ( 15). This yields the following simple expressions for the matrix element of the strong anomaly over the vacuum and rl(rl’):
17) = (ns) cos 13- ITO) sine,
We thus have three parameters to fix: fo, j”s and 0. Two of them can be determined from the radiative widths of v and v’, which are given by
Jr]‘) = 1~~) sine + 1~~)cos e.
(14)
Traditional SUF (3) -based analyses suggest either 0 N -10’ or B N -23” from the quadratic and linear versions, respectively, of the Gell-Mann-Okubo mass formula, whereas we leave 8 as an open parameter, at least for the time being. Analogously to the pion decay constant f,,., Eq. (4), we define the decay constants * fs and fe as couplings of 7s and 70, respectively, to the divergence of the relevant axial-vector currents (where fs = fr # fo in the SUF( 3) limit):
(17)
m;
I(77--r2y)=----(y 961r”
r(7f --f2y)
rn$ 2 = cy 96~’
sine -+ f8
(19) As in Ref. [ I ] we solve these relations for the decay constants and thus get fo and fg as functions of the mixing angle. As experimental input we use [9] I(q + 2~) = (0.51 f 0.026) keV, I( 7’ --f 2~) = (4.53 & 0.59) keV,
where G$, is the gluonic field-strength tensor and z;* =&2 fi,,pr,GApaits dual. Defining as in Ref. [ 1 ] thzYinterpolating fields of r] and 77’as linear combinations of the axial-vector divergences, we thus have (OlJ,A~j~)
-m~.fscos&
(OlJ,A,“Iq)
=-mtfcsin8,
(0 I c?,A~ ( 7’) = rn$ fg sine, (Ola,ArI+)
=mi,focosO.
(16)
where according to the suggestion in the full listings part of the Review of Particle Properties (Ref. [ 91, p. 145 1) we have only retained the more recent data. In Fig. 1, fo/fTand fs/ ffl are plotted as functions of the mixing angle (solid lines) together with their experimental error (dashed lines). We find thus fg/fT2 1 for 0 less than N”- 15’ as expected from chiral perturbation theory. The precise value [ 111 fs/fr= 1.25, however, corresponds to a rather large’ mixing 0 = -21.3”. 9 In fact, Ref.
fo
[ 121
supplements the more traditional approaches
to the 2y decays of 7 and 7 contributions,
x Note that by virtue of the “hard” non-conservation of A:,
(20)
by taking into account continuum
which just cancel the effect of the large S&(3)
I .25.
breaking ratio fg/fT
=
is scale-dependent as described in detail in the first reference of
with the experimental
input data of Eq. (20).
The result is then 19= -(
17 f
2)’
[2]; we will assume here that the momentum dependence can be
the following, this value is in excellent agreement with the results
neglected.
we find from the investigation of other decay channels.
As we shall see in
P. Ball et al./ Physics Letters B 365 (1996) 367-376
371
Table 2 Theoretical formulas and experimental values of the on-shell V-q, 7’ electromagnetic vertex couplings defined in Eq. ( 12). In Fig. 2 the theoretical values are plotted as functions of 0. We use 8v - 40.3’ for the 4-w mixing angle. Experimental values are taken from [9] P
V
77
P
gvpy in units -
W
gwy (exp.)
fvp2
-A
4fx
cosH -
J
J
31,
- -
8 fo
3 I -costi + J5 sin0 4fu 8 fo
P
(I.31 &O.l2)GeV-’
rl’
w
cos 0 sin 0 sin Hv (v5cos.9~ - sinfIv) - ___ 4fx 2&I cos 8 sin#v + @ (JzcosBv - sin&)
‘1
4J
--
‘I’
4
-
w
4fu
Mfo
cos 0 4fx
cos 0
2dfo
(cos&~ + &sin&)
COSSV- !!Z
4fu
2.5 2 1.5 1 0.5 -25
-20
-15
-10
-5
sin 0 - 2~zfo
cos&
(cost+ + &sin&)
3
0
( 1.85 i 0.34) GeV-’
sin0
0
I9 Fig. I. fo/fr and fg/fv,determined from q -+ 2y and 8’ --t 2y, as functions of the mixing angle. Having thus fixed fc and fs, we continue in the next section with the electromagnetic properties of 7 and v’, i.e. the vector meson decays, consider then in Section 5 their light quark matrix elements, to finish in Section 6 with their glue content.
4. We are now in a position to deal with the radiative vector meson decays involving 7 or 7’. In Table 2 we list the decay channels together with the theoretical formulas for the couplings go+ and their experimental values. In Fig. 2 we then plot the theoretical and experimental values as functions of the pseudoscalar mixing angle. Since in our approach continuum contri-
(0.60fO.l5)GeV(0.45 L!Z 0.06) GeV-’ (0.70 f 0.03) GeV-’
< 1.85 GeV-’
butions are neglected, the couplings should be slightly overestimated by about 10 to 15%, as one may also infer from the cleanest channel w --t 7ry investigated in Section 2 (cf. Table 1). A look at Fig. 2(b) shows that this is not the case for 7’ --f py. In that channel, however, we expect a contamination of the experimental value from non-resonant VJTfinal states, which are produced by a completely different mechanism, which is related to the box-anomaly diagram (cf. e.g. [ 131). For p + vy and o --) r]y, Figs. 2(a) and (c), the experimental errors are too large to allow any serious restriction on 8. Within the error bars, however, our predictions agree with the experimental values. gwtljy, on the other hand, is very sensitive to the 4-w mixing angle; for instance, reducing 8~ by 1.5” brings the coupling within 20% agreement with the experimental data, i.e. just what we would expect. As seen from the figures, the most stringent constraint on 0 currently stems from the decay I$ -+ qy. It is largely independent of the 4-w mixing angle and, as it will turn out below, only marginally consistent with the range of mixing angles found from other processes (remember, however, that we use 1 (+ experimental errors, and do not take into account theoretical uncertainties). The issue should be clarified when experimental errors eventually shrink, and in particular we look forward to a measurement of the 4 ---f $y mode, for which our prediction is only a factor 2 below the current experimental upper bound.
P. Ball et al./Physics
312
Letters B 365 (1996) 367-376
(4
(b) 1.6
-i
2
El.
F
cn
2.2 2 1.8 1.6 1.4 1.2 1
I I I I(/
I,,,,,,,,,,,,
I,,,
/,,,I
_ d $
1.5
5
1.3
1.4
=; 1.2 cn 1.1 -25 -20 -15 -10 -5 8
0
.-‘_ .’ .-
_._._._._.-._._._._.-.r’-._._., ----_____.---
.-.-
._._._.-.
- .-.-._._._.
1 kl I
_.__,“._ .-
F _‘-__
_.---__ )
I/
.--
--____.----
I
I
I
I 1
I
I
I I I 1
I,
1 , 1
I
(C) 0.80 0.75 0.70 0.65 0.60 0.55 0.50 0.45 0.40
I I
, I I I I,,
I I I,
I,
I I{
I
//
Id
-25 -20 -15 -10 -5 0
0 (d)
I / , , I
.-.-.-.-._._._._._.-.-.~.~.~
.-.-.-._._._._._._.-.-.-._.-
I1
._._._._._.-._._____.-.-._._ /II I I I I,, II1 I I I I I I
/ /
I
I I I
-25 -20 -15 -10 -5
I
0
“~‘1”~“1’~‘~“~‘~~11”“’
-25 -20 -15 -10 -5 0
0 (f)
1 0.9
0.8 0.7 0.6 0.5
-25 -20 -15 -10 -5
0
-25 -20 -15 -10 -5
0
0 Fig. 2. Theoretical and experimental values of the V-q, 7’ electromagnetic couplings as functions of the pseudoscalar mixing angle. Solid lines: theoretical predictions according to Table 2. Short-dashed lines: errors of the theoretical couplings coming from the experimental errors of fo and fs. Dashed-dotted lines: experimental couplings including errors (for g4,r,r, p lotted in (f), there exists only an experimental upper bound).
313
P. Ball et al. /Physics L.&fen B 365 (1996) 367-376
5. Let us now turn to the investigation of other matrix elements of 7 and q’ that can be probed in our approach. We start with the light quark matrix elements, which determine the decay 77 --t 37~.As in Ref. [ 1] we find lo : 1 focos0 - JZfssin8 (“‘W5u”)=
x
- Jz
fifocos8+fssin8
5
0.6
: 3-
0.5
7
0.4
g
0.3
k
0.2
(21)
(Ol~Y54V)~
-_-_-_-_-_~_-.~_ _._._._._.~._ b .-.-___-._(_-
x f~cos*t? -f:
4-I
\
11”“““““’ -25 -20
1,
-15
.
‘b”l”““‘lq -10 -5
_$ 0
Fig. 3. Theoretical (solid line) and experimental values (dashed-dotted lines, including experimental error) of the decay rate T(r) + 3?r”) as function of the pseudoscalar mixing angle. The short dashes give the theoretical error of the rate, which is dominated by the error of the quark mass ratio.
mi(m, - 3m,)*&(E!$)* focos8-JZfssint9 ( JzfocosB
* *
0
I-(?? + 37rO)
45
_L-.-.-.-.-.-.-
1
which yields the decay rate
=s
_C-.a.-.-_-.-.-.
, \ \
+ fg sin0 >
* ’
(22)
with a kinematical factor S,, = 0.86. The most crucial ingredient in that formula is the ratio of quark masses r = (md - m,)/m,. There exist several possibilities to fix r from next-to-leading order calculations in chiral perturbation theory, as discussed in [ 141, for example. One of them is based essentially on the analysis of r] + 37r in Ref. [ 151. Since in that study 71is necessarily treated as a Goldstone boson, the value of r obtained that way is inconsistent with our analysis. For similar reasons, we also refrain from taking into account r determined from r( (//’ + J/h) /r( I,Q --+ J/@r) . We thus rely on the determination of r from pseudoscalar mass relations, which according to Ref. [ 141 yield r = 0.030 f 0.005. In Fig. 3 we plot the decay rate as a function of 8. Although the agreement with the experimental rate is very good for 0 x -17“, the theoretical errors are large and dominated by the uncertainty in the quark mass ratio. Actually the true theoretical error may be even larger and the rate enhanced by final state interactions [ 14,151. In Ref. [ I] we had also considered the decay 7’ -+ 31r. We have, however, since argued [ 161 that this channel is considerably more complex and also receives contributions from the leading decay mode 7’ ----f~TTT through 9-r mixing. While this study is of great interest in itself and while a measurement of loEq. (21 ) differs slightly from the corresponding Eq. ( 19) in Ref. [ I], where we had made the simplification fo/fs N i
the charged mode v’ 4 ~T+T-T’ could shed considerable light on the mechanism underlying 7’ --t ~ITV, it is of little help in determining the value of 8. 6. We finally turn to the most important issue concerning the 7-77’ system, namely the glue connection. Following the suggestion of Novikov et al., Ref. [ 171, the glue matrix elements are tested in the ratio of the decay rates r(J/$ -+ v(+)y):
=
(O/G&') (OIG~IT)
*
(1-+/m:/,)” (1 -mG/m:/,)”
(23)
Before turning to numerics, let us first comment shortly on the theoretical accuracy of the above equation. Actually the decay rates were calculated in exactly the same approximation we used in deriving the gvh given in the tables, namely dominance of the ground state and neglection of continuum contributions to the dispersion relations. It was also pointed out in Ref. [ 181 that the decay mechanism via the strong anomaly dominates only due to the smallness of the c quark mass and may be non-effective already in Y decays, for which unfortunately no experimental data exist to date. It is for these reasons that we prefer to consider the ratio RJI, instead of the two decay rates separately: both radiative and continuum corrections are expected to cancel to some extent in RJ/$.
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P. Ball et al. / Physics Letters B 365 (I 996) 367-376
these initial fields. While various procedures are possible (like introducing some “glue” Q in the definition of v’, as we did above when using the total divergences as interpolating fields), the choice of basis is essentially free, and the authors of Ref. [ 21 preferred to redefine the Q field, introducing G as below ” . This leads to (we follow closely the notation of [ 2 ] ):
-25
-20
-15
-10
-5
0
0
G=Q-
Fig. 4. RJI$ = F( J/q5 -+ $y)/T( J/I+ ---) 7~) as function of the mixing angle 8. Solid line: RJ/, according to Eqs. ( 17) and (23) including experimental errors of fu and fs (short dashes). Long-dashed line: the ndive prediction (24). Dashed-dotted lines: experimental value with errors.
In Fig. 4 we plot R,/+ according to Eqs. (17) and (23) and also the naive prediction
0-
m~+&d3
RJl* =Cot2* (1 _ m;/m;,,)”
(24)
as functions of the mixing angle. As experimental input we use RT$ = 5.0 f 0.6 [9]. Our prediction (23) is rather sensitive to the mixing angle. If we estimate the remaining theoretical uncertainty in (23) as N lo%, we obtain 0 = -( 17.3 f 1.3)‘. It is clearly seen, on the other hand, that the naive curve is less consistent with the data and requires a very large mixing angle. To understand the departure of our formula from the ndive prediction, we briefly evoke the elegant, but heavier formulation by Veneziano and collaborators [2]. These authors deal with a simplified model, where r]s-ve mixing is neglected. To apply their approach to the physical situation requires the introduction of three initial fields (instead of two in our approach) :
(25) The definition of interpolating fields corresponding to physical particles requires the orthonormalisation of
(26)
where the coefficients bi and Wij can be determined in terms of two-point functions. Expressing the relevant Ward-Takahashi identities in terms of the effective action and taking two derivatives with respect to the gluon fields (g, with indices omitted below), we obtain for the (non-anomalous) octet axial current
=0
(27)
with X a computable coefficient. Clearly, if the proper vertex between G and two gluons could be neglected (in the way the similar coupling between G and two photons is legitimately neglected in Ref. [ 2]), we would recover the ndive result (24). This step is, however, not permitted since the proper vertex between C (whatever its physical realization) and two gluons is precisely expected to be large. It should thus not be surprising that we have to depart from Eq. (24), which is difficult to reconcile with the data, as already noted by Gilman and Kauffman in Ref. [ IO]. We will not delve deeper into this approach, except to mention that it is a clear generalization of ours (in our case the weight of glue in 7’ is directly fixed by our definition of its interpolating field, instead of resulting from a further diagonalization between the three states). 7. Having a rather comprehensive view of the 17 and 77’system and its connection with glue, we now address other promising channels for future investigations, to wit D,y decays. Here two processes compete ‘I Note that the question if G is to be identified with any physical state, a glueball for instance, is left open in Ref. [21.
P. Bail et al./ Physics Letters B 365 (19%) 367-376
to produce 7;1 (v’) X final states: the first one is the decay c + s, leading to an ss pair, which hadronizes to r] or v’, while the second one can be described as CS annihilation into W and two gluons. It is suppressed by the Zweig rule, but may in the present case gain importance through the large (0 )GG (q(7'))matrix elements. We should thus expect both an enhancement of the (~7+ 7$)X modes (unless the interference is destructive) and a large q//v ratio, once the rates are corrected for phase-space and final state interactions. If we ignore for the moment final state interactions, the simplest way to evaluate a possible enhancement is by direct comparison to similar DO decays. For instance the partial width for DO -+ K-p+ (to which only the first diagram contributes) would, in the absence of the glue contribution, be expected to be of the same order (after phase-space corrections) as the --) q’p. Instead we find sumD,-+r]p+D,
375
to be enhanced by an extra factor of up to three when compared with the corresponding DO decay amplitudes, even after including final state interactions; and the global fit done in Ref. 1211 for all non-leptonic D and D, decays including final state interactions, W exchange and annihilation contributions fails to explain the large branching ratios of D, into ~7’71and q’p. The enhancement needed is, however, well in line with our expectations from the extra Zweig-suppressed (gluonmediated) amplitudes. Very recently, the CLEO collaboration has measured the theoretically cleanest semileptonic channel and finds [22] r( DS + T’ev) = 0.35 f 0.09 f 0.07. I(D, ---fTV)
(32)
Defining the relevant form factors as (rlI~~,cID.~)=f:(q~)(~o,+~~),
I(DS -+ %+)(Pq)-3 -tI(D, + 71P)(P71)-3 I(Do --f K-p) (PK)-3
+ fL(#)
2
x!z?i
( mDC,)
=
4.2 f 1.O,
(28)
while for the branching ratios we find =4.5& 1.5,
(29)
which indicates both an enhancement of the decay rate and a large ratio for these important decay modes (of the order of 10% each). The situation is unfortunately less clear in the T associated modes, where the comparison with DO now gives + T/T) (P,~) -1 + ua
w. = 1.6 f 0.4,
---) K-T)
--) VT) (P,)-’
m-l
(30)
while (31) More sophisticated analyses of non-leptonic D, decays were the subject of several studies, e.g. [ 19-211, which, however, mostly relied on the ndive picture of v-$ mixing without taking into account the anomaly. In Ref. [ 201 it was pointed out that in order to account for the data the non-leptonic rates involving 7’ need
(33)
with q = pD, - p,, and analogously J$, CLEO’s measurement yields
f$(O) = 1.14*0.17f0.13, f:(o)
-
-3
r(D,
(PO, - P1I)p )
(34)
where we assumed monopole form factors I2 with a pole at q’ = mZ,.. The ndive mixing model, on the other hand, predikts cos8 - &sin0 fh2) = fP(q2> sin0 + Jicosd’
(35)
which translates into 8 = -( 13.5 f 4.7)’ and is only marginally consistent with the determination from 77-+ 2Y [121. As a temporary conclusion to this section, we would like to add that, while this sector is clearly difficult to investigate both theoretically and experimentally, it could prove very fruitful for our understanding of the role of anomalies and the importance of gluons. There is also the strong suggestion that D, decays, not unlike the J/$ radiative decays, constitute a “gluerich” channel, where search for possible glueball states ‘*We
do not expect the anomalous contribution to change drasti-
cally the y* dependence of the form factors, which in D -+ Kev is a monopole experimentally.
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should be considered. We hope to come back to that issue in future work. Let us finally remark that apart from understanding the structure of the 77 and 7’ particles, the size of their connection to glue is of importance to investigate possible glueball candidates [23]. In particular, the candidates f0 ( 1500) from Crystal Barrel [ 241 and fc ( 1590) [ 251 are seen to decay more frequently into vr]’ than into 77~ (after phase-space correction), a ratio which we expect to be given by
where pt$,, are the respective centre-of-mass momenta. This relation is, however, highly momentum dependent, since we are close to the threshold, and obviously needs to be integrated over the particle width. In conclusion, the above study gives a reasonably consistent description of the 11-v system for values of the mixing angle /3between -20 and - 17 degrees, taking into account anomalies, as tested from various channels, respectively depicting eletromagnetic properties, light quark and gluon content. Our approach will be tested by a general improvement of the experimental measurements, including the V -+ Py and P -+ Vy rates, and in particular by the observation of 4 --+ r]‘y. The investigation of “glue-rich” channels, namely the traditional radiative decays of J/J/, but also, as advocated here, the D,$ decays, will also yield a better understanding of this system and at the same time help in the search for glueballs, which might solve the puzzle. We thank C. Amsler, M. Benayoun, X.Y. Pham, M. Shifman and G. Veneziano for interesting discussions. P.B. gratefully acknowledges the hospitality of the LPTHE of Universite Paris VII, where part of this work was done. J.-M.F. also acknowledges discussions with 1.1.Bigi. This work was supported in part by the European Network Flavourdynamics (ref. chrx-ct930132) and NATO grant (ref. CRG920611).
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